For hydrogen atom, energy of nth level is given by

Options

A. `E_n=-(13.6/n)eV`
B. `E_n=+(13.6/n)eV`
C. `E_n=(13.6/n^2)eV`
D. `E_n=-(13.6/n^2)eV`

Answer

To find the energy of the nth level of a hydrogen atom, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Formula**: The energy of the nth level for hydrogen-like atoms is given by the formula:
\[ E_n = -\frac{m z^2 e^4}{8 \epsilon_0^2 h^2 n^2} \]
where:
- \( E_n \) is the energy of the nth level,
- \( m \) is the mass of the electron,
- \( z \) is the atomic number (number of protons),
- \( e \) is the charge of the electron,
- \( \epsilon_0 \) is the permittivity of free space,
- \( h \) is Planck's constant,
- \( n \) is the principal quantum number.

2. **Substitute Values for Hydrogen Atom**: For a hydrogen atom, \( z = 1 \). Therefore, the formula simplifies to:
\[ E_n = -\frac{m e^4}{8 \epsilon_0^2 h^2 n^2} \]

3. **Insert Known Constants**: We will now substitute the known values:
- Mass of electron \( m = 9.1 \times 10^{-31} \, \text{kg} \)
- Charge of electron \( e = 1.6 \times 10^{-19} \, \text{C} \)
- Permittivity of free space \( \epsilon_0 = 8.85 \times 10^{-12} \, \text{F/m} \)
- Planck's constant \( h = 6.63 \times 10^{-34} \, \text{Js} \)

4. **Calculate the Energy**: Substitute these values into the formula:
\[ E_n = -\frac{(9.1 \times 10^{-31}) (1.6 \times 10^{-19})^4}{8 (8.85 \times 10^{-12})^2 (6.63 \times 10^{-34})^2 n^2} \]

5. **Simplify the Expression**: After performing the calculations, we find that:
\[ E_n \approx -\frac{2.2 \times 10^{-18}}{n^2} \, \text{J} \]

6. **Convert to Electron Volts**: To convert joules to electron volts, we use the conversion factor \( 1 \, \text{eV} = 1.6 \times 10^{-19} \, \text{J} \):
\[ E_n = \frac{-2.2 \times 10^{-18}}{1.6 \times 10^{-19} n^2} \, \text{eV} \approx -\frac{13.6}{n^2} \, \text{eV} \]

7. **Final Result**: Therefore, the energy of the nth level of a hydrogen atom is:
\[ E_n = -\frac{13.6}{n^2} \, \text{eV} \]